Convert a polynomial to hypergeometric function

In summary, there is no known method for converting a series of polynomials into a hypergeometric function. The only way to obtain the solution is through direct verification.
  • #1
azizianhra
3
0
i want to write a hypergeometric function (2F1(a,b;c,x)) as function of n that generate polynomials below

n=0 → 1
n=1 → y
n=2 → 4(ω+1)y^2-1
n=3 → y(2(2ω+3)y^2-3)
n=4 → 8(ω+2)(2ω+3)y^4-6(6+4ω)y^2+3
... → ...


2F1(a,b;c,x)=1+(ab)/(c)x+(a(a+1)b(b+1))/(c(c+1))x^2/2!+...


the answer is 2F1(-n,n+4ω+2;2ω+3/2;(1-y)/2) but i want the solution completely.
 
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  • #2
azizianhra said:
i want to write a hypergeometric function (2F1(a,b;c,x)) as function of n that generate polynomials below

n=0 → 1
n=1 → y
n=2 → 4(ω+1)y^2-1
n=3 → y(2(2ω+3)y^2-3)
n=4 → 8(ω+2)(2ω+3)y^4-6(6+4ω)y^2+3
... → ...


2F1(a,b;c,x)=1+(ab)/(c)x+(a(a+1)b(b+1))/(c(c+1))x^2/2!+...


the answer is 2F1(-n,n+4ω+2;2ω+3/2;(1-y)/2) but i want the solution completely.

What is preventing you from verifying the results for n = 0, 1, 2, 3, 4 for yourself? You can work out the solution just as easily as we can.
 
  • #3
Ray Vickson said:
What is preventing you from verifying the results for n = 0, 1, 2, 3, 4 for yourself? You can work out the solution just as easily as we can.

i want to proof it. if i don't know the answer how can i yield a,b,c and x in hypergeometric function?
 
  • #4
azizianhra said:
i want to proof it. if i don't know the answer how can i yield a,b,c and x in hypergeometric function?

You told us that the answer is 2F1(-n,n+4ω+2;2ω+3/2;(1-y)/2), so you are told what must be a, b and c. You prove it (not "proof" it) by straight verification.
 
  • #5
I need a method of solution that starts form series of polynomials and convert it to hypergeometric function. i mean how to achieving answer that we already have it. I know that by substituting n in answer the polynomials generated, but this isn't the point that i need it. I'm looking for a analytical solution.
 
  • #6
azizianhra said:
I need a method of solution that starts form series of polynomials and convert it to hypergeometric function. i mean how to achieving answer that we already have it. I know that by substituting n in answer the polynomials generated, but this isn't the point that i need it. I'm looking for a analytical solution.

Good luck with that; I don't think there is any reasonable and general method.
 

FAQ: Convert a polynomial to hypergeometric function

1. What is a polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, combined using addition, subtraction, and multiplication. It can also include exponents, but not division or square roots.

2. What is a hypergeometric function?

A hypergeometric function is a special type of mathematical function that is defined by a hypergeometric series. It is used to solve equations and can be expressed as a ratio of two polynomials.

3. How do you convert a polynomial to a hypergeometric function?

To convert a polynomial to a hypergeometric function, you need to identify the variables and coefficients in the polynomial and then rewrite it in the form of a hypergeometric series. This can be done using mathematical formulas and techniques.

4. What is the purpose of converting a polynomial to a hypergeometric function?

The conversion of a polynomial to a hypergeometric function allows for a more efficient and compact representation of the polynomial. It also allows for the use of hypergeometric identities and techniques to solve mathematical problems and equations.

5. Can any polynomial be converted to a hypergeometric function?

No, not all polynomials can be converted to hypergeometric functions. The polynomial must have specific properties and coefficients in order to be converted, and the conversion process may not always be possible or practical.

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